Explicit Linear Kernels for Packing Problems

Abstract

During the last years, several algorithmic meta-theorems have appeared (Bodlaender et al. [FOCS 2009], Fomin et al. [SODA 2010], Kim et al. [ICALP 2013]) guaranteeing the existence of linear kernels on sparse graphs for problems satisfying some generic conditions. The drawback of such general results is that it is usually not clear how to derive from them constructive kernels with reasonably low explicit constants. To fill this gap, we recently presented [STACS 2014] a framework to obtain explicit linear kernels for some families of problems whose solutions can be certified by a subset of vertices. In this article we enhance our framework to deal with packing problems, that is, problems whose solutions can be certified by collections of subgraphs of the input graph satisfying certain properties. ${\mathcal F}$-Packing is a typical example: for a family ${\mathcal F}$ of connected graphs that we assume to contain at least one planar graph, the task is to decide whether a graph G contains k vertex-disjoint subgraphs such that each of them contains a graph in ${{\mathcal {F}}}$ as a minor. We provide explicit linear kernels on sparse graphs for the following two orthogonal generalizations of ${{\mathcal {F}}}$-Packing: for an integer $\ell \geqslant 1$, one aims at finding either minor-models that are pairwise at distance at least $\ell$ in G ($\ell$-$\mathcal {F}$-Packing), or such that each vertex in G belongs to at most $\ell$ minors-models ($\mathcal {F}$-Packing with$\ell$-Membership). Finally, we also provide linear kernels for the versions of these problems where one wants to pack subgraphs instead of minors.

Deferred Proofs in Sect. 3

1.1 Proof of Lemma 1

Let us first show that the equivalence relation $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} ^*$ has finite index. Let $I \subseteq \{1,\ldots ,t\}$. Since we assume that $\mathcal { E}_{\mathcal {F}{SP}}^{}$ is g-confined, we have that for any $G \in \mathcal {B}_t$ with $\Lambda (G)=I$, the function $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G,\ \cdot \ )$ can take at most $g(t)+2$ distinct values ($g(t)+1$ finite values and possibly the value $-\infty$). Therefore, it follows that the number of equivalence classes of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} ^*$ containing all graphs $G \in \mathcal {B}_t$ with $\Lambda (G)=I$ is at most ${(g(t)+2)^{|\mathcal {C}^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (I)|}}$. As the number of subsets of $\{1,\ldots ,t\}$ is $2^t$, we deduce that the overall number of equivalence classes of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} ^*$ is at most ${(g(t)+2)^{s_{\mathcal { E}_{\mathcal {F}{SP}}^{}}(t)}} \cdot 2^t$. Finally, since the equivalence relation $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^*$ is the Cartesian product of the equivalence relations $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} ^*$ and $\sim _{\mathcal {G},t}$, the result follows from the fact that $\mathcal {G}$ can be expressed in MSO logic.

1.2 Proof of Fact 1

Let $G = G^- \oplus G_B$ and let $G' = G^- \oplus G_B'$. Assume that $G \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} ^* G'$. In order to deduce that $G \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^* G'$, it suffices to prove that $G \sim _{\mathcal {G},t} G'$. Let $H\in \mathcal {B}_t$. We need to show that $G \oplus H \in \mathcal {G}$ if and only if $G' \oplus H \in \mathcal {G}$. We have that $G \oplus H = (G_B \oplus G^-) \oplus H = G_B \oplus (G^- \oplus H)$, and similarly for $G'$. Since $G_B \sim _{\mathcal {G},t} G_B'$, it follows that $G \oplus H = G_B \oplus (G^- \oplus H) \in \mathcal {G}$ if and only if $G_B \oplus (G^- \oplus H) = G \oplus H \in \mathcal {G}$.

1.3 Proof of Lemma 2

Let $\mathcal { E}_{\mathcal {F}{SP}}^{} = (\mathcal {C}^{\mathcal { E}_{\mathcal {F}{SP}}^{}},L^{\mathcal { E}_{\mathcal {F}{SP}}^{}},f^{\mathcal { E}_{\mathcal {F}{SP}}^{}})$ be a $\Pi$-encoder and let $G_1,G_2 \in \mathcal {B}_t$ such that $G_1 \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} G_2$. We need to prove that for any $H\in \mathcal {B}_t$ and any integer k, $(G_1 \oplus H, k ) \in \Pi$ if and only if $(G_2 \oplus H, k + \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2)) \in \Pi$.

Suppose that $(G_1 \oplus H, k ) \in \Pi$ (by symmetry the same arguments apply starting with $G_2$). Since $G_1 \oplus H$ is a 0-boundaried graph and $\mathcal { E}_{\mathcal {F}{SP}}^{}$ is a $\Pi$-encoder, we have that

$$\begin{aligned} f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_1 \oplus H,R_{\emptyset }) = f^{\Pi }(G_1 \oplus H) \geqslant k. \end{aligned}$$

(9)

As $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^*$ is DP-friendly and $G_1 \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^* G_2$, it follows that $(G_1 \oplus H) \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^* (G_2 \oplus H)$ and that $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1 \oplus H,G_2 \oplus H) = \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2)$. Since $G_2 \oplus H$ is also a 0-boundaried graph, the latter property and Eq. (9) imply that

$$\begin{aligned} f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_2 \oplus H,R_\emptyset ) = f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_1 \oplus H,R_\emptyset ) + \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2) \geqslant k + \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2). \end{aligned}$$

(10)

Since $\mathcal { E}_{\mathcal {F}{SP}}^{}$ is a $\Pi$-encoder, $f^\Pi (G_2 \oplus H) = f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_2 \oplus H,R_\emptyset )$, and from Eq. (10) it follows that $(G_2 \oplus H, k + \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2)) \in \Pi$.

1.4 Proof of Lemma 3

Let ${\mathfrak {C}}$ be an arbitrary equivalence class of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$, and let $G_1,G_2 \in {\mathfrak {C}}$. Let us first argue that ${\mathfrak {C}}$ contains some progressive representative. Since $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2) = f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_1,R) -f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_2,R)$ for every encoding R such that $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_1,R),f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G_2,R) \ne - \infty$, $G \in {\mathfrak {C}}$ is progressive if $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G,R)$ is minimal in $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} ({\mathfrak {C}},R)= \{f(G,R) : G \in {\mathfrak {C}} \}$ for every encoding R (including those for which the value is $- \infty$). Since $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} ({\mathfrak {C}},R)$ is a subset of ${\mathbb {N}} \cup \{ - \infty \}$, it necessarily has a minimal element, hence there is a progressive representative in ${\mathfrak {C}}$ (in other words, the order defined by $G_1 \preccurlyeq G_2$ if $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2)\leqslant 0$ is well-founded).

Now let $G \in \mathcal {G}$ be a progressive representative of ${\mathfrak {C}}$ with minimum number of vertices. We claim that G has size at most $2^{r({\mathcal {E}},g,t,{\mathcal {G}})+1} \cdot t$ (we would like to stress that at this stage we only need to care about the existence of such representative G, and not about how to compute it). Let $(T,\mathcal {X})$ be a boundaried nice tree decomposition of G of width at most $t-1$ such that $\partial (G)$ is contained in the root-bag (such a nice tree decomposition exists by [36]).

We first claim that for any node x of T, the graph $G_x$ is a progressive representative of its equivalence class with respect to $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$, namely $\mathfrak {C'}$. Indeed, assume for contradiction that $G_x$ is not progressive, and therefore we know that there exists $G_x' \in \mathfrak {C'}$ such that $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_x',G_x) < 0$. Let $G'$ be the graph obtained from G by replacing $G_x$ with $G_x'$. Since $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^*$ is DP-friendly, it follows that $G \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} G'$ and that $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G',G) = \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_x',G_x) < 0$, contradicting the fact that G is a progressive representative of the equivalence class ${\mathfrak {C}}$.

We now claim that for any two nodes $x,y \in V(T)$ lying on a path from the root to a leaf of T, it holds that $G_x \not \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} G_y$. Indeed, assume for contradiction that there are two nodes $x,y \in V(T)$ lying on a path from the root to a leaf of T such that $G_x \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} G_y$. Let $\mathfrak {C'}$ be the equivalence class of $G_x$ and $G_y$ with respect to $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$. By the previous claim, it follows that both $G_x$ and $G_y$ are progressive representatives of $\mathfrak {C'}$, and therefore it holds that $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_y,G_x) = 0$. Suppose without loss of generality that $G_y \subsetneq G_x$ (that is, $G_y$ is a strict subgraph of $G_x$), and let $G'$ be the graph obtained from G by replacing $G_x$ with $G_y$. Again, since $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^*$ is DP-friendly, it follows that $G \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} G'$ and that $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G',G) = \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_y,G_x) = 0$. Therefore, $G'$ is a progressive representative of ${\mathfrak {C}}$ with $|V(G')| < |V(G)|$, contradicting the minimality of |V(G)|.

Finally, since for any two nodes $x,y \in V(T)$ lying on a path from the root to a leaf of T we have that $G_x \not \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} G_y$, it follows that the height of T is at most the number of equivalence classes of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$, which is at most $r( \mathcal { E}_{\mathcal {F}{SP}}^{},g,t,\mathcal {G})$ by Lemma 1. Since T is a binary tree, we have that $|V(T)| \leqslant 2^{r({\mathcal {E}},g,t,{\mathcal {G}})+1} - 1$. Finally, since $|V(G)| \leqslant |V(T)| \cdot t$, it follows that $|V(G)| \leqslant 2^{r({\mathcal {E}},g,t,{\mathcal {G}})+1} \cdot t$, as we wanted to prove.

1.5 Proof of Lemma 4

Let $\mathcal { E}_{\mathcal {F}{SP}}^{} = (\mathcal {C}^{\mathcal { E}_{\mathcal {F}{SP}}^{}},L^{\mathcal { E}_{\mathcal {F}{SP}}^{}},f^{\mathcal { E}_{\mathcal {F}{SP}}^{}})$ be the given encoder. We start by generating a repository ${\mathfrak {R}}$ containing all the graphs in $\mathcal {F}_t$ with at most $b+1$ vertices. Such a set of graphs, as well as a boundaried nice tree decomposition of width at most $t-1$ of each of them, can be clearly generated in time depending only on b and t. By assumption, the size of a smallest progressive representative of any equivalence class of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$ is at most b, so ${\mathfrak {R}}$ contains a progressive representative of any equivalence class of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$ with at most b vertices. We now partition the graphs in ${\mathfrak {R}}$ into equivalence classes of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$ as follows: for each graph $G \in {\mathfrak {R}}$ and each encoding $R \in \mathcal {C}^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (\Lambda (G))$, as $L^{\mathcal { E}_{\mathcal {F}{SP}}^{}}$ and $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}}$ are computable, we can compute the value $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (G,R)$ in time depending only on $\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,$ and b. Therefore, for any two graphs $G_1,G_2 \in {\mathfrak {R}}$, we can decide in time depending only on $\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,b$, and $\mathcal {G}$ whether $G_1 \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} G_2$, and if this is the case, we can compute the transposition constant $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (G_1,G_2)$ within the same running time.

Given a t-protrusion Y on n vertices with boundary $\partial (Y)$, we first compute a boundaried nice tree decomposition $(T,\mathcal {X},r)$ of Y in time $f(t) \cdot n$, by using the linear-time algorithm of Bodlaender [7, 36]. Such a t-protrusion Y equipped with a tree decomposition can be naturally seen as a t-boundaried graph by assigning distinct labels from $\{1,\ldots ,t\}$ to the vertices in the root-bag. We can assume that $\Lambda (Y)=\{1,\ldots ,t\}$. Note that the labels can be transferred to the vertices in all the bags of $(T,\mathcal {X},r)$, by performing a standard shifting procedure when a vertex is introduced or removed from the nice tree decomposition [8]. Therefore, each node $x \in V(T)$ defines in a natural way a t-protrusion $Y_x \subseteq Y$ with its associated boundaried nice tree decomposition, with all the boundary vertices contained in the root bag. Let us now proceed to the description of the replacement algorithm.

We process the bags of $(T,\mathcal {X})$ in a bottom-up way until we encounter the first node x in V(T) such that $|V(Y_x)|=b+1$ (note that as $(T,\mathcal {X})$ is a nice tree decomposition, when processing the bags in a bottom-up way, at most one new vertex is introduced at every step, and recall that $b \geqslant t$, hence such an x exists). We compute the equivalence class ${\mathfrak {C}}$ of $Y_x$ according to $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$; this corresponds to computing the set of encodings $\mathcal {C}^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (\Lambda (Y_x))$ and the associated values of $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}} (Y_x,\cdot )$ that, by definition of an encoder, can be calculated since $f^{\mathcal { E}_{\mathcal {F}{SP}}^{}}$ is a computable function. As $|V(Y_x)|=b+1$, the graph $Y_x$ is contained in the repository ${\mathfrak {R}}$, so in constant time we can find in ${\mathfrak {R}}$ a progressive representative $Y_x'$ of ${\mathfrak {C}}$ with at most b vertices and the corresponding transposition constant $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Y_x',Y_x) \leqslant 0$, (the inequality holds because $Y_x'$ is progressive). Let Z be the graph obtained from Y by replacing $Y_x$ with $Y_x'$, so we have that $|V(Y)| < |V(Z)|$ (note that this replacement operation directly yields a boundaried nice tree decomposition of width at most $t-1$ of Z). Since $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^*$ is DP-friendly, it follows that $Y \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} Z$ and that $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Z,Y) = \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Y_x',Y_x) \leqslant 0$.

We recursively apply this replacement procedure on the resulting graph until we eventually obtain a t-protrusion $Y'$ with at most b vertices such that $Y \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} Y'$. The corresponding transposition constant $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Y',Y)$ can be easily computed by summing up all the transposition constants given by each of the performed replacements. Since each of these replacements introduces a progressive representative, we have that $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Y',Y) \leqslant 0$. As we can assume that the total number of nodes in a nice tree decomposition of Y is O(n) [36, Lemma 13.1.2], the overall running time of the algorithm is O(n) (the constant hidden in the “O” notation depends indeed exclusively on $\mathcal { E}_{\mathcal {F}{SP}}^{},g,b,\mathcal {G}$, and t).

1.6 Proof of Theorem 1

By Lemma 1, the number of equivalence classes of the equivalence relation $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$ is finite and by Lemma 3 the size of a smallest progressive representative of any equivalence class of $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t}$ is at most $b(\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,\mathcal {G})$. Therefore, we can apply Lemma 4 and deduce that, in time O(|Y|), we can find a t-protrusion $Y'$ of size at most $b(\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,\mathcal {G})$ such that $Y \sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} Y'$ and the corresponding transposition constant $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Y',Y)$ with $\Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Y',Y) \leqslant 0$. Since $\mathcal { E}_{\mathcal {F}{SP}}^{}$ is a $\Pi$-encoder and $\sim _{\mathcal { E}_{\mathcal {F}{SP}}^{},\mathcal {G},t} ^*$ is DP-friendly, it follows from Lemma 2 that $Y \equiv _{\Pi } Y'$ and that $\Delta _{\Pi ,t} (Y',Y) = \Delta _{\mathcal { E}_{\mathcal {F}{SP}}^{},t} (Y',Y) \leqslant 0$. Therefore, if we set $k' := k +\Delta _{\Pi ,t} (Y',Y)$, it follows that (G, k) and $((G - (Y-\partial (Y)))\oplus Y',k')$ are indeed equivalent instances of $\Pi$ with $k' \leqslant k$ and $|Y'| \leqslant b(\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,\mathcal {G})$.

1.7 Proof of Corollary 1

For $1 \leqslant i \leqslant \ell$, where $\ell$ is the number of protrusions in the decomposition, we apply the polynomial-time algorithm given by Theorem 1 to replace each t-protrusion $Y_i$ with a graph $Y_i'$ of size at most $b(\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,\mathcal {G})$ and to update the parameter accordingly. In this way we obtain an equivalent instance $(G',k')$ such that $G' \in \mathcal {G}$, $k' \leqslant k$ and $|V(G')| \leqslant |Y_0| + \ell \cdot b(\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,\mathcal {G}) \leqslant (1+b(\mathcal { E}_{\mathcal {F}{SP}}^{},g,t,\mathcal {G}))\alpha \cdot k$ .